A new method of focusing x-rays is described using appropriately tapered capillaries. The x-rays are incident on the inner surface of the capillary below the critical glancing angle and reflect due to total external reflection. By appropriately narrowing the capillary, the x-rays can thus be focused in a broad band of energies. The theory of the effect and optimum taper is described. A measurement verifying the focusing capability of the method is presented. The method appears practical for focusing bending magnet synchrotron radiation around 2 keV down to a diameter of 10 mu m from an initial dimension 1 mm2 incident cross section with an attenuation of the total energy of about 2, i.e., an increase in the intensity per unit area of 6.5 x 103. Greater focusing is possible with softer x-rays and from undulator sources. The technique has wide-ranging applicability. This proposal requests funding to develop this new methodology and to apply it in conjunction with flow methods in order to obtain kinetic x-ray absorption fine structure (XAFS) data of excited molecular states and dynamic protein conformational states with um sec time resolution.